Adaptive soft-output detector for magnetic tape read channels

ABSTRACT

In accordance with one embodiment, a computer program product includes a computer readable storage medium having computer readable program code that is readable and/or executable by a processor to: receive a signal including precoded data read from a magnetic tape medium and pass the signal through a soft detector to calculator first soft information about each bit of the signal and to provide adaptive compensation for the precoded data, send the first soft information to a soft decoder, pass the signal through the soft decoder to calculate second soft information about each bit the signal, and send the second soft information to the soft detector, wherein the precoded data is passed through at least one precoder prior to being written to the magnetic tape medium via a precoder that applies 1/(1⊕D 2 ) to bits of data, where D is delay corresponding to bit duration

RELATED APPLICATIONS

This application is a continuation of copending U.S. patent applicationSer. No. 13/527,508, filed Jun. 19, 2012, which is herein incorporatedby reference.

BACKGROUND

The present invention relates to data storage using magnetic tapechannels, and more particularly, to reading data using an adaptivesoft-output detector in the magnetic tape read channels.

A conventional data flow 200 for reading data from a magnetic tape isshown in FIG. 2, according to the prior art. The tape channel 202receives data that has passed through an error correction code (ECC)encoder and a modulation code (MC) encoder prior to being stored tomagnetic tape. Then, the tape channel 202 reads the data as bits x_(k)(either a 0 or a 1), noise n_(k) is injected into the signal to become asignal y_(k) that is read by the hard detector 204. This detectionutilizes a hard detector 204, which does not incorporate softinformation in its detecting scheme.

Soft information may be considered a probability that a detected bit (0or 1) is actually a 0 or a 1. There are different types of probabilitiesthat may be implemented in a data flow, but any data flow which usesprobabilities (soft information) for decoding must utilize softdetection. However, soft detection in magnetic tape recording channelshave typically suffered from some problems. Two types of soft detectiontypically used are BCJR, which is named after its inventors, Bahl,Cocke, Jelinek, and Raviv, and Dual-Max, which is a derivative(simplified version) of BCJR that relies on a specific set ofassumptions.

However, each of these algorithms experience issues when implemented indata flows for magnetic tape recording channels. Accordingly, it wouldbe beneficial to have a soft detector that is capable of operating inmagnetic tape recording channel data flow that alleviates the issuesassociated with known algorithms.

BRIEF SUMMARY

In one embodiment, a data storage system includes a soft detectorconfigured to calculate first soft information about each bit of asignal including precoded data read from a magnetic tape medium via atape channel, and send the first soft information to a soft decoder,wherein the precoded data includes a characteristic of being passedthrough at least one precoder prior to being written to the magnetictape medium, wherein the soft detector provides automatic compensationfor the precoded data, wherein the soft detector applies a Dual-Max(DMAX) algorithm including:

${\alpha_{k}\left( S_{k} \right)} \cong {\max\limits_{S_{k - 1}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\alpha_{k - 1}\left( S_{k - 1} \right)}} \right\}}$${\beta_{k - 1}\left( S_{k - 1} \right)} \cong {\max\limits_{S_{k}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}}$$\begin{matrix}{{{LLR}\left( a_{k} \right)} \cong {{\max\limits_{\underset{a_{k} = {+ 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}} -}} \\{{\max\limits_{\underset{a_{k} = {- 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}},}\end{matrix}$wherein y_(k) is the signal, α_(k) denotes a bit in a bit sequence ofthe signal, α_(k)(S_(k)) is an alpha term for a current state (S_(k)) ina forward recursion, α_(k)(S_(k−1)) is an alpha term for a previousstate (S_(k−1)) in the forward recursion β_(k)(S_(k)) is a beta term forthe current state in a backward recursion, β_(k−1)(S_(k−1)) is a betaterm for the previous state in the backward recursion, and LLR(α_(k)) isan approximation of a log-likelihood term that calculates a posterioriprobabilities.

In another embodiment, a method includes reading precoded data from amagnetic tape medium to produce a signal, passing the signal through anoise whitening filter, wherein the noise whitening filter is configuredto control variance of noise affecting the signal output from thefilter, wherein the noise whitening filter processes the signalaccording to the following transfer polynomial: W(D)=1−(p₁D+ . . .+p_(λ)D^(λ)), where p₁ . . . p_(λ) are the noise whitening coefficientswith the noise whitening filter having 2^(λ) states, and where the tapechannel is characterized by a transfer polynomial F(D)=1+f₁D+ . . .+f_(L)D^(L) where D is delay corresponding to bit duration, with thetape channel having 2^(λ) states, L represents a memory length of thetape channel, and λ represents a memory length of the noise whiteningfilter, passing the signal through a soft detector to calculate firstsoft information about each bit of the signal and to provide adaptivecompensation for the precoded data, sending the first soft informationto a soft decoder, passing the signal through the soft decoder tocalculate second soft information about each bit of the signal, andsending the second soft information to the soft detector, wherein theprecoded data includes a characteristic of being passed through at leastone precoder prior to being written to the magnetic tape medium.

In yet another embodiment, a computer program product includes acomputer readable storage medium having computer readable program codeembodied therewith, the computer readable program code beingreadable/executable by a processor to cause the processor to: receive,by the processor, a signal including precoded data read from a magnetictape medium and pass the signal through a soft detector to calculatefirst soft information about each bit of the signal and to provideadaptive compensation for the precoded data, send, by the processor, thefirst soft information to a soft decoder, pass, by the processor, thesignal through the soft decoder to calculate second soft informationabout each bit of the signal, and send, by the processor, the secondsoft information to the soft detector, wherein the precoded dataincludes a characteristic of being passed through at least one precoderprior to being written to the magnetic tape medium, and wherein thecharacteristic of the precoded data is consistent with a precoder thatapplies 1/(1⊕D²) to bits of data b_(k), where D is delay correspondingto bit duration.

Any of these embodiments may be implemented in a magnetic data storagesystem such as a tape drive system, which may include a magnetic head, adrive mechanism for passing a magnetic medium (e.g., recording tape)over the magnetic head, and a controller electrically coupled to themagnetic head.

Other aspects and embodiments of the present invention will becomeapparent from the following detailed description, which, when taken inconjunction with the drawings, illustrates by way of example theprinciples of the invention.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 illustrates a simplified tape drive of a tape-based data storagesystem, according to one embodiment.

FIG. 2 shows a data flow for a magnetic tape recording channel,according to the prior art.

FIG. 3 is a partial data flow for BCJR detection, according to the priorart.

FIG. 4 is a partial reverse concatenation architecture with softdetection and soft decoding, according to one embodiment.

FIG. 5 is a system for exchanging soft information for a magnetic taperecording channel, according to one embodiment.

FIG. 6 shows Viterbi-algorithm-like computation on a channel trellisforward and backward in time, according to one embodiment.

FIG. 7 shows one embodiment of a soft detection/decoding system using anoise predictive soft Dual-MAX (DMAX), according to one embodiment.

FIG. 8 shows an adaptive whitening filter coefficient updater, accordingto one embodiment.

FIG. 9 shows a soft detection/decoding system using a softnoise-predictive data-dependent (DD) DMAX detector, according to oneembodiment.

FIG. 10 shows an adaptive whitening filter coefficient updater,according to one embodiment.

FIG. 11 shows one embodiment of a soft detection/decoding system whichuses a soft detector that provides adaptive compensation for thepresence of a precoded tape channel, according to one embodiment.

FIG. 12 shows one embodiment of a precoded tape channel.

FIG. 13 shows one embodiment of a soft detection/decoding system whichuses a soft detector that provides adaptive compensation for thepresence of a precoded tape channel, according to one embodiment.

FIG. 14 shows one embodiment of a precoded tape channel.

FIG. 15 shows examples of trellis structures according to variousembodiments.

DETAILED DESCRIPTION

The following description is made for the purpose of illustrating thegeneral principles of the present invention and is not meant to limitthe inventive concepts claimed herein. Further, particular featuresdescribed herein can be used in combination with other describedfeatures in each of the various possible combinations and permutations.

Unless otherwise specifically defined herein, all terms are to be giventheir broadest possible interpretation including meanings implied fromthe specification as well as meanings understood by those skilled in theart and/or as defined in dictionaries, treatises, etc.

It must also be noted that, as used in the specification and theappended claims, the singular forms “a,” “an,” and “the” include pluralreferents unless otherwise specified.

In one general embodiment, a data storage system includes a tape channelfor reading precoded data from a magnetic tape medium to produce asignal, a soft detector adapted for calculating first soft informationabout each bit of the signal and sending the first soft information to asoft decoder, and the soft decoder positioned subsequent to the softdetector, the soft decoder being adapted for calculating second softinformation about each bit of the signal and sending the second softinformation to the soft detector, wherein the precoded data includes acharacteristic of being passed through at least one precoder prior tobeing written to the magnetic tape medium, and wherein the soft detectorprovides automatic compensation for the precoded data.

In another general embodiment, a method includes reading precoded datafrom a magnetic tape medium to produce a signal, passing the signalthrough a soft detector to calculate first soft information about eachbit of the signal and to provide adaptive compensation for the precodeddata, sending the first soft information to a soft decoder, passing thesignal through the soft decoder to calculate second soft informationabout each bit of the signal, and sending the second soft information tothe soft detector, wherein the precoded data includes a characteristicof being passed through at least one precoder prior to being written tothe magnetic tape medium.

In yet another general embodiment, a computer program product includes acomputer readable storage medium having computer readable program codeembodied therewith, the computer readable program code includingcomputer readable program code configured for reading precoded data froma magnetic tape medium to produce a signal, computer readable programcode configured for passing the signal through a soft detector tocalculate first soft information about each bit of the signal and toprovide adaptive compensation for the precoded data, computer readableprogram code configured for sending the first soft information to a softdecoder, computer readable program code configured for passing thesignal through the soft decoder to calculate second soft informationabout each bit of the signal, and computer readable program codeconfigured for sending the second soft information to the soft detector,wherein the precoded data includes a characteristic of being passedthrough at least one precoder prior to being written to the magnetictape medium.

As will be appreciated by one skilled in the art, aspects of the presentinvention may be embodied as a system, method or computer programproduct. Accordingly, aspects of the present invention may take the formof an entirely hardware embodiment, an entirely software embodiment(including firmware, resident software, micro-code, etc.) or anembodiment combining software and hardware aspects that may allgenerally be referred to herein as “logic.” a “circuit.” a “module,” ora “system.” Furthermore, aspects of the present invention may take theform of a computer program product embodied in one or more computerreadable medium(s) having computer readable program code embodiedthereon.

Any combination of one or more computer readable medium(s) may beutilized. The computer readable medium may be a computer readable signalmedium or a non-transitory computer readable storage medium. Anon-transitory computer readable storage medium may be, for example, butnot limited to, an electronic, magnetic, optical, electromagnetic,infrared, or semiconductor system, apparatus, or device, or any suitablecombination of the foregoing. More specific examples (a non-exhaustivelist) of the computer readable storage medium would include thefollowing: a portable computer diskette, a hard disk, a random accessmemory (RAM), a read-only memory (ROM), an erasable programmableread-only memory (EPROM or Flash memory), a portable compact discread-only memory (CD-ROM), an optical storage device, a magnetic storagedevice, or any suitable combination of the foregoing. In the context ofthis document, a computer readable storage medium may be anynon-transitory, tangible medium that can contain, or store a program foruse by or in connection with an instruction execution system, apparatus,or device.

A computer readable signal medium may include a propagated data signalwith computer readable program code embodied therein, for example, inbaseband or as part of a carrier wave. Such a propagated signal may takeany of a variety of forms, including, but not limited to,electro-magnetic, optical, or any suitable combination thereof. Acomputer readable signal medium may be any computer readable medium thatis not a computer readable storage medium and that can communicate,propagate, or transport a program for use by or in connection with aninstruction execution system, apparatus, or device, such as anelectrical connection having one or more wires, an optical fiber, etc.

Program code embodied on a computer readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wireline, optical fiber cable, RF, etc., or any suitable combination ofthe foregoing.

Computer program code for carrying out operations for aspects of thepresent invention may be written in any combination of one or moreprogramming languages, including an object oriented programming languagesuch as Java, Smalltalk, C++ or the like and conventional proceduralprogramming languages, such as the “C” programming language or similarprogramming languages. The program code may execute entirely on theuser's computer, partly on the user's computer, as a stand-alonesoftware package, partly on the user's computer and partly on a remotecomputer or entirely on the remote computer or server. In the latterscenario, the remote computer may be connected to the user's computerthrough any type of network, including a local area network (LAN) or awide area network (WAN), or the connection may be made to an externalcomputer (for example, through the Internet using an Internet ServiceProvider).

Aspects of the present invention are described below with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems) and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer program instructions. These computer program instructions maybe provided to a processor of a general purpose computer, specialpurpose computer, or other programmable data processing apparatus toproduce a machine, such that the instructions, which execute via theprocessor of the computer or other programmable data processingapparatus, create means for implementing the functions/acts specified inthe flowchart and/or block diagram block or blocks.

These computer program instructions may also be stored in a computerreadable medium that can direct a computer, other programmable dataprocessing apparatus, or other devices to function in a particularmanner, such that the instructions stored in the computer readablemedium produce an article of manufacture including instructions whichimplement the function/act specified in the flowchart and/or blockdiagram block or blocks.

The computer program instructions may also be loaded onto a computer,other programmable data processing apparatus, or other devices to causea series of operational steps to be performed on the computer, otherprogrammable apparatus or other devices to produce a computerimplemented process such that the instructions which execute on thecomputer or other programmable apparatus provide processes forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks.

FIG. 1 illustrates a simplified tape drive 100 of a tape-based datastorage system, which may be employed according to various embodiments.While one specific implementation of a tape drive is shown in FIG. 1, itshould be noted that the embodiments described herein may be implementedin the context of any type of tape drive system.

As shown, a tape supply cartridge 120 and a take-up reel 121 areprovided to support a tape 122. One or more of the reels may form partof a removable cassette and are not necessarily part of the system 100.The tape drive, such as that illustrated in FIG. 1, may further includedrive motor(s) to drive the tape supply cartridge 120 and the take-upreel 121 to move the tape 122 over a tape head 126 of any type. The tape122 may be a linear tape open (LTO) format or any other suitablemagnetic tape medium known in the art.

Guides 125 guide the tape 122 across the tape head 126. Such tape head126 is in turn coupled to a controller assembly 128 via a cable 130. Thecontroller 128 typically comprises a servo channel and controls headfunctions, such as track following, writing, reading, etc. The cable 130may include read/write circuits to transmit data to the head 126 to berecorded on the tape 122 and to receive data read by the head 126 fromthe tape 122. An actuator 132 determines position of the head 126relative to the tape 122.

An interface 134 may also be provided for communication between the tapedrive and a host (integral or external) to send and receive the data andfor controlling the operation of the tape drive and communicating thestatus of the tape drive to the host, all as will be understood by thoseof skill in the art.

BCJR soft detection relies on the α posteriori probability (APP) ofstates and state transitions in a channel finite-state machine, andadheres to the flow shown in FIG. 3. The ISI channel 302 is described bya trellis diagram with state S_(k)ε{0, 1, . . . , 2^(L)−1} at time k.The joint probability density p(S_(k−1), S_(k), Y₁ ^(N)) plays a centralrole in the algorithm because of the following relationship, where thesummation occurs over all trellis branches for which α_(k)=+1 (as shown)and −1.

$\left. {P\left( {a_{k} = \left. {+ 1} \middle| Y_{1}^{N} \right.} \right)} \right.\sim{\sum\limits_{\underset{a_{k} = {+ 1}}{S_{k - 1}->S_{k}}}{p\left( {S_{k - 1},S_{k},Y_{1}^{N}} \right)}}$

The BCJR detector 304 factors a joint probability density as follows,where the first term is computed through forward recursion, the secondterm is the branch transition probability, and the third term iscomputed through backward recursion.

⁢p ⁡ ( S k - 1 , S k , Y 1 N ) = ⁢ p ⁡ ( S k - 1 , S k , Y 1 k - 1 , y k , Yk + 1 N ︵ ) = ⁢ p ⁡ ( S k - 1 , Y 1 k - 1 ) · p ⁡ ( y k , S k | S k - 1 )· p ⁡ ( Y k + 1 N | S k ) = ⁢ p ⁡ ( S k - 1 , Y 1 k - 1 ) ︸ α _ k - 1 ⁡ ( Sk - 1 ) · p ⁢ ( y k | S k - 1 , S k ) · P ⁡ ( S k | S k - 1 ) ︸ γ _ k ⁡ (S k - 1 , S k ) · ⁢ p ⁡ ( Y k + 1 N | S k ) ︸ β _ k ⁡ ( S k ) ⁢ Forward ⁢ ⁢Recursion ⁢ : ⁢ ⁢ α _ ⁡ ( S k ) = ∑ S k - 1 ⁢ γ _ k ⁡ ( S k - 1 , S k ) · α _k - 1 ⁡ ( S k - 1 ) Branch ⁢ ⁢ Transition ⁢ ⁢ Probability ⁢ : ⁢ ⁢ p ⁢ ( y k | Sk - 1 , S k ) = 1 2 ⁢ π ⁢ σ ⁢ ⅇ - ( y k - x k ) 2 / 2 ⁢ ⁢ σ 2 P ⁡ ( S k | Sk - 1 ) = P ⁡ ( a k ) ⁢ Reverse ⁢ ⁢ Recursion ⁢ : ⁢ ⁢ β _ k - 1 ⁡ ( S k - 1 ) =∑ S k ⁢ γ _ k ⁡ ( S k - 1 , S k ) · β _ k ⁡ ( S k )

The BCJR algorithm uses a maximum α posteriori (MAP) detection rule,where the binary value â_(k) is selected that yields the larger APPaccording to either the likelihood ratio (LR) or the log likelihoodratio (LLR), as follows.

$\begin{matrix}{{{LR}\left( a_{k} \right)} = \frac{P\left( {a_{k} = \left. {+ 1} \middle| Y_{1}^{N} \right.} \right)}{P\left( {a_{k} = \left. {- 1} \middle| Y_{1}^{N} \right.} \right)}} & \longrightarrow & \left\{ \begin{matrix}{\left. {{{LR}\left( a_{k} \right)} \geq 1}\Rightarrow{\hat{a}}_{k} \right. = {+ 1}} \\{\left. {{{LR}\left( a_{k} \right)} < 1}\Rightarrow{\hat{a}}_{k} \right. = {- 1}}\end{matrix} \right. \\{{{LLR}\left( a_{k} \right)} = {\ln\frac{P\left( {a_{k} = \left. {+ 1} \middle| Y_{1}^{N} \right.} \right)}{P\left( {a_{k} = \left. {- 1} \middle| Y_{1}^{N} \right.} \right)}}} & \longrightarrow & \left\{ \begin{matrix}{\left. {{{LLR}\left( a_{k} \right)} \geq 0}\Rightarrow{\hat{a}}_{k} \right. = {+ 1}} \\{\left. {{{LLR}\left( a_{k} \right)} < 0}\Rightarrow{\hat{a}}_{k} \right. = {- 1}}\end{matrix} \right.\end{matrix}$

The BCJR may be advantageously formulated in the log domain, such thatthe following relationships are created.

$\begin{matrix}{{\overset{\_}{\alpha}}_{k}\left( S_{k} \right)} & & {{\alpha_{k}\left( S_{k} \right)} = {\ln\left\lbrack {{\overset{\_}{\alpha}}_{k}\left( S_{k} \right)} \right\rbrack}} \\{{\overset{\_}{\beta}}_{k}\left( S_{k} \right)} & & {{\beta_{k}\left( S_{k} \right)} = {\ln\left\lbrack {{\overset{\_}{\beta}}_{k}\left( S_{k} \right)} \right\rbrack}} \\{{\overset{\_}{\gamma}}_{k}\left( {S_{k - 1},S_{k}} \right)} & & {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} = {\ln\left\lbrack {{\overset{\_}{\gamma}}_{k}\left( {S_{k - 1},S_{k}} \right)} \right\rbrack}} \\\begin{matrix}{{p\left( {S_{k - 1},S_{k},Y_{1}^{N}} \right)} =} \\{{{\overset{\_}{\alpha}}_{k - 1}\left( S_{k - 1} \right)}{{\overset{\_}{\gamma}}_{k}\left( {S_{k - 1},S_{k}} \right)}{{\overset{\_}{\beta}}_{k}\left( S_{k} \right)}}\end{matrix} & & \begin{matrix}{{p\left( {S_{k - 1},S_{k},Y_{1}^{N}} \right)} =} \\{\mathbb{e}}^{{\alpha_{k - 1}{(S_{k - 1})}} + {\gamma_{k}{({S_{k - 1},S_{k}})}} + {\beta_{k}{(S_{k})}}}\end{matrix} \\{{{LR}\left( a_{k} \right)} = \frac{P\left( {a_{k} = \left. {+ 1} \middle| Y_{1}^{N} \right.} \right)}{P\left( {a_{k} = \left. {- 1} \middle| Y_{1}^{N} \right.} \right)}} & & {{{LLR}\left( a_{k} \right)} = {\ln\frac{P\left( {a_{k} = \left. {+ 1} \middle| Y_{1}^{N} \right.} \right)}{P\left( {a_{k} = \left. {- 1} \middle| Y_{1}^{N} \right.} \right)}}}\end{matrix}$

Another algorithm that may be used in soft detection is called aDual-Max algorithm, which is based on BCJR with some modifications andsimplifications. A simplified (sub-optimum) algorithm is obtained basedon the following approximation.

${\ln\left\lbrack {\sum\limits_{i}{\mathbb{e}}^{\delta_{i}}} \right\rbrack} \cong {\max\limits_{i}\delta_{i}}$

Then, the max-Log-MAP (Dual-Max or DMAX) algorithm may be modeled asfollows:

${\alpha_{k}\left( S_{k} \right)} \cong {\max\limits_{S_{k - 1}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\alpha_{k - 1}\left( S_{k - 1} \right)}} \right\}}$${\beta_{k - 1}\left( S_{k - 1} \right)} \cong {\max\limits_{S_{k}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}}$$\begin{matrix}{{{LLR}\left( \alpha_{k} \right)} \cong {{\max\limits_{\underset{a_{k} = {+ 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}} -}} \\{\max\limits_{\underset{a_{k} = {- 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}}\end{matrix}$

A partial reverse concatenation architecture 400 with soft detection 402and soft decoding 404 is shown in FIG. 4, according to one embodiment.In this architecture, C2 encoding is performed on data, then MCencoding, and then C1 encoding prior to being written to a magnetic tapemedium. Then, in reading data from the tape via the tape channel, signalx_(k) is read with added noise n_(k) to yield the signal y_(k) which isthen sent to the soft detector 402. The soft detector calculates andsends soft information about the bit sequence {α_(k)}406 to the soft C1decoder 404. The soft C1 decoder 404 then calculates and sends softinformation about the bit sequence {α_(k)}408 back to the soft detector402, thereby creating an iterative loop. As the number of iterationsincreases, so does the detection accuracy of the bits in the bitsequence {α_(k)}. In one embodiment, the soft detector 402 may have2^(L) states, as determined by the number of states of the tape channel.

A system 500 for exchanging soft information for a magnetic taperecording channel may be as shown in FIG. 5, according to oneembodiment. In this embodiment, a soft detector 402 (which may be a DMAXdetector) provides a log likelihood ratio (LLR) to calculate aposteriori probabilities (APPs) 502 along with each read bit to a softdecoder 404 (which may be a low-density parity check (LPDC) decoder)which in turn provides the LLR to calculate α priori probabilities 504of the bytes to the soft detector 402. Other types of soft detectors andsoft decoders may be used as known in the art.

As indicated above, the detection operations involve a forwardcomputation step and a backward computation step. These steps can bethought of as corresponding to running a Viterbi-algorithm-likecomputation on a channel trellis forward and backward in time. Oneexample is illustrated in FIG. 6, according to one embodiment.

In FIG. 6, it is assumed that a codeword having a block of N signalsamples y₁, . . . , y_(N) is received at the soft detector. Upperdiagram 602 shows a part of the algorithm that is applied forward intime assuming, for illustrative purposes, a simple 4-state trellis;state values at initial time 0 are denoted by S₀ and state values atfinal time N are denoted by S_(N). The forward algorithm computes a setof values denoted by α₀, α₁, . . . , α_(N). Middle diagram 604 shows thebackward pass, where the quantities β_(N), . . . , β₂, β₁ are computedon the same trellis. Lower diagram 606 shows that by combining thevalues obtained in the forward and backward passes (α₀, α₁, . . . ,α_(N) and β₀, β₁, . . . , β_(N)), it is possible to compute softinformation (which is shown as a log-likelihood ratio LLR) on theindividual bits α_(k) that form the codeword being processed. These LLRvalues may then be passed to the soft decoder, in one embodiment. Ofcourse, in other approaches, other soft information may be calculated,such as other likelihood calculations, as would be known to one of skillin the art.

In the embodiments described in FIGS. 7-10, the soft detector uses aDMAX detection algorithm. Additionally, for the methods described belowaccording to various embodiments, it is assumed that the DMAX detectoris used at least twice: once in a first initial pass after receivinginput signal samples where detection does not include using softinformation from the soft decoder; and another time in at least a secondpass where soft information provided by the soft decoder is used todetect the block of signal samples.

Referring now to FIG. 7, one embodiment of a soft detection/decodingsystem 700 using a noise predictive soft DMAX detector 702 is shown.After bits of data are read from the tape via the tape channel 202,where x_(k)=α_(k)+f₁α_(k−1)+ . . . +f_(Lαk−L), the signal x_(k) affectedby noise n_(k) (which may be colored noise in some approaches) isdenoted by y_(k) (which equals x_(k)+n_(k)). Signal y_(k) is thensubsequently input to a noise whitening filter 704. The output z_(k) ofthe noise whitening filter 704 is then input to the soft detector 402which provides a LLR to calculate APPs along with each read bit (shownas output 406) to the soft decoder 404. The soft decoder 404 nextcalculates a LLR for each bit to provide the APPs of the bits to thesoft detector 402. In one embodiment, the soft detector 402 may be aDMAX soft detector, as described herein in more detail.

In one embodiment, a noise predictive soft DMAX detector 702 (whichincludes the noise whitening filter 704 and the soft detector 402) mayhave 2^(L+λ) states, where L and λ represent the memory length of thechannel 202 and the noise whitening filter 704, respectively. This isbecause each of the L and λ memory units corresponding to the channeland the whitening filter, respectively, are capable of storing a binaryvalue of 0 or 1, and accordingly the noise predictive soft DMAX detector702 may have a total of 2^(L+λ) states.

The noise whitening filter 704 attempts to minimize the variance of thenoise affecting the signal z_(k) that is input to the detector 702. Inone approach, the whitening filter 704 may apply the following transferfunction (polynomial) to y_(k): W(D)=1−(p₁D+ . . . +p_(λ)D^(λ)), where Drepresents delay corresponding to a bit duration, and the tape channel202 itself is characterized by a transfer polynomial F(D)=1+f₁D+ . . .+f_(L)D^(L).

The noise variance term now refers to the noise as seen at the output ofthe whitening filter 704, and the branch metric m_(k)(S_(k−1), S_(k))used by the detector 702, according to one embodiment, may berepresented by:

${m_{k}\left( {S_{k - 1},S_{k}} \right)} = {{- \frac{\left( {z_{k} - w_{k}} \right)^{2}}{2\sigma_{p}^{2}}} + {\ln\;{P\left( a_{k} \right)}}}$with the prediction noise variance being σp², w_(k) being an idealnominal signal associated with transition from state S_(k−1) to stateS_(k), and z_(k) being the actual output of the noise whitening filter.

In more embodiments, more than one whitening filter 704 may be employed.For example, 2, 4, 8, 16, 32, 64, or more whitening filters 704 may beemployed, such as in a bank of whitening filters. There may be Mwhitening filters 704 in the bank of whitening filters. In one approach,M may equal the number of branches of the soft detector 402 trellis,e.g., a number of state transitions of the soft detector 402, such as2^(L+λ+1) filters.

According to another embodiment, the noise-predictive soft DMAX detector702 may have normalization applied thereto. When the DMAX algorithm isnormalized, the α (forward) and β (backward) variables may berepresented with a predetermined finite number of bits as opposed to thevalues of these variables growing in magnitude without bound, as ispossible using conventional DMAX algorithms. In addition, andadvantageously, normalization does not affect LLR values. Onenormalization operation is shown below:

${\alpha_{k}\left( S_{k} \right)} = {\max\limits_{S_{k - 1}}{\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)}} \right\}\mspace{14mu}\left( {{k = 1},\ldots\mspace{14mu},N} \right)}}$${{{Normalization}\mspace{11mu}\text{:}\mspace{14mu}{find}\mspace{14mu} A_{k}} = {\max\limits_{S_{k}}\left\{ {\alpha_{k}\left( S_{k} \right)} \right\}}},{{{replace}\mspace{14mu}{\alpha_{k}\left( S_{k} \right)}}->{{\alpha_{k}\left( S_{k} \right)} - {A_{k}\mspace{14mu}{\forall{\alpha_{k}\left( S_{k} \right)}}}}}$${\beta_{k - 1}\left( S_{k - 1} \right)} = {\max\limits_{S_{k}}{\left\{ {{\beta_{k}\left( S_{k} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)}} \right\}\mspace{14mu}\left( {{k = N},\ldots\mspace{14mu},2} \right)}}$${{{Normalization}\mspace{11mu}\text{:}\mspace{14mu}{find}\mspace{14mu} B_{k - 1}} = {\max\limits_{S_{k - 1}}\left\{ {\beta_{k - 1}\left( S_{k - 1} \right)} \right\}}},{{{replace}\mspace{14mu}{\beta_{k - 1}\left( S_{k - 1} \right)}}->{{\beta_{k - 1}\left( S_{k - 1} \right)} - {B_{k - 1}\mspace{14mu}{\forall{\beta_{k - 1}\left( S_{k - 1} \right)}}}}}$Of course, other algorithms for normalization may be possible as well,according to other embodiments.

In another embodiment, reduced state detection may be performed. Aspreviously indicated, the soft detector 402 may have 2^(L+λ) states. Itis possible to use a longer whitening filter 704 and still keep 2^(L+λ)states in the detector trellis using reduced-state detection. In oneapproach, the whitening filter 704 may apply the following transferpolynomial to y_(k): W(D)=1−(p₁D+ . . . +λ′D^(λ′)), where λ′ is greaterthan λ and assuming that the tape channel 202 itself is characterized bya transfer polynomial F(D)=1+f₁D+ . . . +f_(L)D^(L).

In this case, the bits defining states S_(k−1) and S_(k) are not enoughto specify the bit string of length L+λ+1 needed to compute the systemoutput z_(k). Therefore, the missing bit(s) are taken from the pathmemory associated with each state S_(k−1). The branch metricm_(k)(S_(k−1), S_(k)) may then be written as follows:

${m_{k}\left( {S_{k - 1},S_{k}} \right)} = {{- \frac{\left\{ {z_{k} - {w_{k}\left( S_{k - 1} \right)}} \right\}^{2}}{2\sigma_{p}^{2}}} + {\ln\;{P\left( a_{k} \right)}}}$where the notation w_(k)(S_(k−1)) indicates that the bit pattern thatdefines w_(k), which is an ideal nominal signal associated withtransition from state S_(k−1) to state S_(k), also depends on the pathmemory associated with the previous state S_(k−1), the prediction noisevariance is σ_(p) ², and z_(k) is the actual output of the noisewhitening filter.

In one embodiment, the whitening filter coefficients p₁, p₂, . . . ,p_(λ), may be estimated adaptively as shown in FIG. 8, a simplifiedversion of which is described in U.S. Pat. No. 8,077,764, which isherein incorporated by reference. Referring again to FIG. 8, thewhitening filter coefficients p₁, p₂, . . . , p_(λ) may be updated usinga whitening filter coefficient updater 802 as follows (λ=2 is shown forsimplicity in FIG. 8):p ₁ ←p ₁ +αe _(k) {circumflex over (n)} _(k−1)p ₂ ←p ₂ +αe _(k) {circumflex over (n)} _(k−2)where α is the adaptation stepsize, e_(k) is the error signal, and{circumflex over (n)}_(k−1), {circumflex over (n)}_(k−2) are the noiseestimates at previous time instants. The corresponding whiteningfunction may be defined as follows:W(D)=1−(p ₁ D+p ₂ D ²)

Furthermore, the error signal e_(k) may be input to a prediction noisevariance computation 804 where prediction noise variance σ_(p) ² iscomputed based on the error signal e_(k), and a small number ε (e.g.,0.001). In the embodiment shown in FIG. 8, the last calculatedprediction noise variance σp² is used to calculate the next (updated)prediction noise variance σp².

However, the noise which exists at the output of the tape channel is notjust colored noise, but may also include data dependent noise, which iscommon in magnetic recording. Accordingly, the soft detector may bedevised to take into account this data dependent noise, and therefore doa better job in detecting the bits in the signal. The data dependentnature of the noise may be taken into account in the detection processto achieve better performance in detecting the bits.

The noise that generally affects magnetic data recording systems iseither electronics noise or medium noise. It is a combination of thesenoise sources that produces noise that is difficult to remove from thesignal. Noise from the electronics may be either white or colored, butit is not data dependant. The medium noise is a data dependent noise:this type of noise is specific to magnetic recording channels because itcorresponds to the written transition in the medium and the positionand/or the width of the transition may be variable. There is nocertainty in how long the transition is, where the transition ispositioned, etc. In addition, if a transition is not written, then thistype of noise will not affect the readback signal from the medium.

This is why medium noise is actually data dependent, because atransition is actually written when there is a transition between bitsand it is only then that medium noise will manifest itself. Toaccommodate for this, instead of having only one noise whitening filteras previously described in relation to FIG. 7, more than one noisewhitening filter may be present, such as two, four, eight, sixteen,etc., and each of these noise whitening filters is conducting filteringbased on a unique possible data pattern being read. For example, thenoise whitening filter coefficients may correspond to the particulardata pattern associated with that particular noise whitening filter. Inone example, if the detector is attempting to determine the likelihoodof α_(k) and α_(k−1), which may be 0 0, 0 1, 1 0, or 1 1, four noisewhitening filters may be used, one dependent on each of the possibledata patterns. For the four noise whitening filter outputs eachcorresponding to one of the four possible data patterns, the likelihoodof each data pattern is computed by the soft detector. For example, thesoft DMAX detector receives outputs from each of the noise whiteningfilters in the bank of noise whitening filters and calculates alikelihood of a data pattern associated with each of the outputs.Accordingly, the filtering is data dependent, since each noise whiteningfilter is tailored to be specific to a particular data pattern.

Referring now to FIG. 9, one embodiment of a soft detection/decodingsystem 900 using a soft noise-predictive data-dependent (DD) DMAXdetector 902 is shown. As shown, multiple whitening filters 704 are usedin a bank of whitening filters 906. However, this is not required, as asingle whitening filter 704 may be used, or any number thereof, e.g., 2,4, 8, 16, 32, 64, or more.

Magnetic recording systems are typically affected by electronics noiseas well as medium noise from the tape medium itself. Medium noiseresults primarily from transition position/width variations encounteredduring recording operations. Since positions and shapes of magnetizationtransitions are determined by the symbols to be written as bits on thetape medium, medium noise depends on the input data sequence, asunderstood by those of skill in the art.

As previously described, the soft DD-DMAX detector 904 may have 2^(L+λ)states, and there are M=2^(L+λ+1) noise whitening filters (that is, onewhitening filter 704 per detector branch). Thus, the characteristics ofthe noise whitening filters 704 enter both the forward and the backwardrecursions on the detector trellis. The prediction noise variance σ_(p)²(α) refers to the noise as seen at the output of each whitening filter704, and the branch metric m_(k)(S_(k−1), S_(k)) may be calculated asfollows:

${m_{k}\left( {S_{k - 1},S_{k}} \right)} = {{{- \frac{1}{2}}\ln\left\{ {\sigma_{p}^{2}\left( \underset{\_}{a} \right)} \right\}} - \frac{\left\{ {{z_{k}\left( \underset{\_}{a} \right)} - {w_{k}\left( \underset{\_}{a} \right)}} \right\}^{2}}{2{\sigma_{p}^{2}\left( \underset{\_}{a} \right)}} + {\ln\;{P\left( a_{k} \right)}}}$where α is the data pattern corresponding to state transitionS_(k−1)→S_(k). In the branch metric equation, w_(k)(α) is an idealnominal signal associated with transition from state S_(k−1) to stateS_(k), and z_(k)(α) is the actual output of the noise whitening filter.

According to another embodiment, the soft noise-predictive DD-DMAXdetector 902 may have normalization applied thereto. When the DMAXalgorithm is normalized, the α (forward) and β (backward) variables maybe represented with a predetermined finite number of bits as opposed tothe values of these variables growing in magnitude without bound, as ispossible using conventional DMAX algorithms. In addition, andadvantageously, normalization does not affect LLR values. Onenormalization operation is shown below:

${\alpha_{k}\left( S_{k} \right)} = {\max\limits_{S_{k - 1}}{\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)}} \right\}\mspace{14mu}\left( {{k = 1},\ldots\mspace{14mu},N} \right)}}$${{{Normalization}\mspace{11mu}\text{:}\mspace{14mu}{find}\mspace{14mu} A_{k}} = {\max\limits_{S_{k}}\left\{ {\alpha_{k}\left( S_{k} \right)} \right\}}},{{{replace}\mspace{14mu}{\alpha_{k}\left( S_{k} \right)}}->{{\alpha_{k}\left( S_{k} \right)} - {A_{k}\mspace{14mu}{\forall{\alpha_{k}\left( S_{k} \right)}}}}}$${\beta_{k - 1}\left( S_{k - 1} \right)} = {\max\limits_{S_{k}}{\left\{ {{\beta_{k}\left( S_{k} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)}} \right\}\mspace{14mu}\left( {{k = N},\ldots\mspace{14mu},2} \right)}}$${{{Normalization}\mspace{11mu}\text{:}\mspace{14mu}{find}\mspace{14mu} B_{k - 1}} = {\max\limits_{S_{k - 1}}\left\{ {\beta_{k - 1}\left( S_{k - 1} \right)} \right\}}},{{{replace}\mspace{14mu}{\beta_{k - 1}\left( S_{k - 1} \right)}}->{{\beta_{k - 1}\left( S_{k - 1} \right)} - {B_{k - 1}\mspace{14mu}{\forall{\beta_{k - 1}\left( S_{k - 1} \right)}}}}}$Of course, other algorithms for normalization may be possible as well,according to other embodiments.

In one embodiment, each noise whitening filter 704 in the noisewhitening filter bank 906 may use a whitening function to minimize thevariance of the noise affecting its output signal. In one approach, thewhitening filter 704 may apply the following transfer polynomial toy_(k): W(D)=1−(p₁D+ . . . +p_(λ)D^(λ)), assuming that the tape channel202 itself is characterized by a transfer polynomial F(D)=1+f₁D+ . . .+f_(L)D^(L).

In another embodiment, the number of noise whitening filters 704 in thenoise whitening filter bank 906 may equal the number of branches in thedetector's trellis structure, e.g., a number of state transitions of thesoft noise-predictive DD-DMAX detector 902. In another embodiment, thenumber of noise whitening filters 704 in the noise whitening filter bank906 may depend on a longer bit pattern than the pattern defined by statetransitions, i.e., there will be more noise whitening filters 704 thanthere are branches on the detector trellis.

Accordingly, the bits defining states S_(k−1) and S_(k) will beinsufficient to match on a one-to-one basis with a whitening filter;therefore, the missing bits may be taken from path memory associatedwith each state S_(k−1). The branch metric m_(k)(S_(k−1), S_(k)) forthis calculation may be written as follows, in one approach:

${m_{k}\left( {S_{k - 1},S_{k}} \right)} = {{{- \frac{1}{2}}\ln\left\{ {\sigma_{p}^{2}\left\lbrack {\underset{\_}{a}\left( S_{k - 1} \right)} \right\rbrack} \right\}} - \frac{\left( {{z_{k}\left\lbrack {\underset{\_}{a}\left( S_{k - 1} \right)} \right\rbrack} - {w_{k}\left\lbrack {\underset{\_}{a}\left( S_{k - 1} \right)} \right\rbrack}} \right)^{2}}{2{\sigma_{p}^{2}\left\lbrack {\underset{\_}{a}\left( S_{k - 1} \right)} \right\rbrack}} + {\ln\;{P\left( a_{k} \right)}}}$where the notation α(S_(k−1)) indicates that the bit pattern αthatdefines a specific whitening filter also depends on the path memoryassociated with the previous state S_(k−1), and where σ_(p) ²[α(S_(k−1))]is data-pattern and previous-state dependent predictionnoise variance.

In yet another embodiment, each noise whitening filter 704 in the noisewhitening filter bank 906 may depend on a shorter bit pattern than thepattern defined by state transitions, i.e., there will be less noisewhitening filters 704 than branches of the detector trellis; therefore,the same noise whitening filter 704 may be used more than once in thedetector trellis.

For example, if in an 8-state detector trellis α is selected asα=(α_(k), α_(k−1), α_(k−2)), then the two distinct branches shown belowmay use the same noise whitening filter 704.S _(k):(α_(k−1),α_(k−2),α_(k−3)=0)→S _(k+1):(α_(k),α_(k−1),α_(k−2))S _(k):(α_(k−1),α_(k−2),α_(k−3)=1)→S _(k+1):(α_(k),α_(k−1),α_(k−2))

In another embodiment, reduced state detection may be performed on anyof the longer, shorter, or equal bit pattern schemes describedpreviously. In reduced state detection, the whitening filter 704 mayextend over a longer time span (i.e., a longer memory), but the softdetector 904 may use the same amount of states as before. As previouslyindicated, the soft detector 904 may have 2^(L+λ) states. It is possibleto use a longer whitening filter 704 and still keep 2^(L+λ) states inthe detector trellis using reduced-state detection. In this case, thebits defining states S_(k−1) and S_(k) are not enough to specify the bitstring of length L+λ+1 needed to compute the system output z_(k). Inthis case, the noise whitening filters may apply the following transferpolynomial: W(D)=1−(p_(i,1)D+ . . . +p_(i,λ′)D^(λ′)), where p_(i,1) . .. p_(i,λ′) are noise whitening coefficients, i is 0 to M−1 where M is atotal number of noise whitening filters in the bank of noise whiteningfilters, the tape channel may be characterized by a transfer polynomialF(D)=1+f₁D+ . . . +f_(L)D^(L), with 2^(L) being a number of states ofthe tape channel, wherein the soft detector has a total of 2^(L+λ)states, and the noise whitening filter ideally (without state reduction)leading to 2^(λ′) states, with λ′ being greater than λ.

Therefore, the missing bit(s) are taken from the path memory associatedwith each state S_(k−1). The branch metric m_(k)(S_(k−1), S_(k)) maythen be written as follows:

${m_{k}\left( {S_{k - 1},S_{k}} \right)} = {{{- \frac{1}{2}}\ln\left\{ {\sigma_{p}^{2}\left\lbrack {\underset{\_}{a}\left( S_{k - 1} \right)} \right\rbrack} \right\}} - \frac{\left( {{z_{k}\left\lbrack {\underset{\_}{a}\left( S_{k - 1} \right)} \right\rbrack} - {w_{k}\left\lbrack {\underset{\_}{a}\left( S_{k - 1} \right)} \right\rbrack}} \right)^{2}}{2{\sigma_{p}^{2}\left\lbrack {\underset{\_}{a}\left( S_{k - 1} \right)} \right\rbrack}} + {\ln\;{P\left( a_{k} \right)}}}$where the notation w_(k)[α(S_(k−1))] indicates that the bit pattern thatdefines w_(k), which is an ideal nominal signal associated withtransition from state S_(k−1) to state S_(k), also depends on the pathmemory associated with the previous state S_(k−1), and where σ_(p)²[α(S_(k−1))] is data-pattern and previous-state dependent predictionnoise variance.

In another embodiment, the whitening filter coefficients p_(i,1),p_(i,2), . . . , p_(i,λ) may be estimated adaptively as shown in FIG.10, a simplified version of which is described in U.S. patentapplication Ser. No. 12/753,586, which is herein incorporated byreference. Referring again to FIG. 10, the whitening filter coefficientsp_(i,1), p_(i,2), . . . , p_(i,λ) may be updated using a whiteningfilter coefficient updater 802 from soft information as follows (in FIG.10, λ=2 is assumed for illustration purposes):p _(i,1) ←p _(i,1) +αe _(i,k) {circumflex over (n)} _(k−1)p _(i,2) ←p _(i,2) +αe _(i,k) {circumflex over (n)} _(k−2)where α is the adaptation stepsize, e_(i,k) is the error signal based onthe address i (in this example, 64 whitening filters 704 are assumed,but any number may be used), and {circumflex over (n)}_(k−1),{circumflex over (n)}_(k−2) are the noise estimates from previous timeinstants, e.g., previous noise estimates. In addition, the whiteningfilter transfer polynomial may be defined as follows:W _(i)(D)=1−(p _(i,1) D+p _(i,2) D ²), i=0, . . . ,63where i=0, . . . , 63 based on 64 whitening filters 704 in the bank ofwhitening filters 906.

Furthermore, the error signal e_(i,k) may be input to a prediction noisevariance computation 804 where prediction noise variance σ_(p) ² iscomputed based on the error signal e_(i,k), and a small number ε (e.g.,0.001). In the embodiment shown in FIG. 10, the prediction noisevariance σ_(p) ²(i) that is used to calculate the next prediction noisevariance σ_(p) ² is input from the bank of variance values 1002. Thenumber of variance values in the bank of variance values 1002 may numberthe same as the number of whitening filters 704 in the bank of whiteningfilters 906, according to one embodiment.

In some instances, the tape may be written by a drive having a precoderpositioned before the C1 encoder, e.g., an LDPC encoder and no precoderpositioned after the encoder and/or immediately adjacent the writechannel. In such case, the detector functions as noted elsewhere herein.However, if the tape was written by a drive with a precoder positionedafter the C1 encoder and/or immediately adjacent the write channel, thenthe system may automatically compensate during signal detection for theeffect on the data caused by the precoder in such position in thewriting device. In either case, the data may have a characteristic ofbeing passed through at least one precoder prior to being written to themagnetic tape medium.

The drive performing data readback may determine if and how the data hasbeen precoded in any of the manners described above and below in amanner known in the art, such as detecting such information from thetape itself, cartridge memory, etc.

Referring now to FIG. 11, there is shown a representation of a tapedrive system 1100 which uses, in a reading section, a soft detector 1102that provides automatic compensation for the presence of precoding inthe tape channel 1104. As shown, in the precoded tape channel 1104, aprecoder 1106 is positioned after a C1 encoder 1108 and immediatelyprior to a tape channel 202, such that the output of the C1 encoder 1108is sent to the precoder 1106, which, in turn, sends the precoder outputto the tape channel 202 for writing on the tape. The tape channel 202 isshown as a single module representing respective components of the readand the write section, as would be understood by one skilled in the art.

The data which passes through the precoder 1106 may be described ashaving a characteristic of being passed through a precoder, such thatthe data may be handled differently and/or compensated for in a detectorwhich reads a tape which has precoded data recorded thereon.

One embodiment of the precoded tape channel 1104 is shown in FIG. 12. Asshown, the algorithms provided by the precoder 1106 and the tape channel202 may be combined, as represented by the precoded tape channel 1104.Referring now to FIGS. 11-12, bits of data, b_(k), are sent to theprecoder 1106. The precoder 1106 may apply the following equation tob_(k): 1/(1⊕²) where D is delay corresponding to a bit duration.

After the output, α_(k), of the precoder 1106 is read from the tape viathe tape channel 202, the noiseless signal may be represented, forexample, as an extended partial-response class 4 (EPR4) signalx_(k)=α_(k)+α_(k−1)−α_(k−2)−α_(k−3). The soft detector 1102 providesautomatic compensation for the presence or absence of a precoder 1106adjacent the write channel in the writing device by accounting for theprecoding function that the precoder 1106 applies to the data before itis written to the tape and clearly also for the transformation that thetape channel 202 applies to the data as it is read from the tape.

For example, in one embodiment, the soft detector 1102 providingautomatic compensation for the presence of a precoded tape channel 1104may calculate the LLR as follows:

$\begin{matrix}{{{LLR}\left( b_{k} \right)} \cong {{\max\limits_{\beta_{1}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}} -}} \\{\max\limits_{\beta_{0}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}}\end{matrix}$where β₀ is the set of all branches in the detector trelliscorresponding to the state transitions S_(k−1)→S_(k) having labelb_(k)=0, and β₁ is the set of all branches in detector trelliscorresponding to the state transitions S_(k−1)−*S_(k) having labelb_(k=1). The branch metric m_(k)(S_(k−1), S_(k)) may then be written asfollows:m _(k)(S _(k−1) ,S _(k))=−(y _(k) −x _(k))²/(2σ²)+ln P(b _(k))where the a priori probabilities P(b_(k)) provided by the soft C1decoder 404 correspond to the appropriate b_(k)-labeled branch beingconsidered (i.e., b_(k=0) or b_(k=)1).

In one approach, the soft detector 1102 providing automatic compensationfor the presence of a precoded tape channel 1104 may be a noisepredictive soft DMAX detector with a noise whitening filter. In anotherapproach, the soft detector 1102 providing automatic compensation forthe presence of a precoded tape channel 1104 may be a softnoise-predictive DD-DMAX detector with one or more noise whiteningfilters. In addition, any of these embodiments may be used in a reducedstate detector, as described in more detail herein.

In accordance with one embodiment, the soft detector 1102 providingautomatic compensation for the presence of a precoded tape channel 1104may retain the same number of states and state transitions (e.g.,branches) as in the case where precoding is not used. Additionally, thesoft detector 1102 providing automatic compensation for the presence ofa precoded tape channel 1104 may retain the same branch labeling withnominal output signal x_(k) as in the case where precoding is not used.

Referring now to FIG. 13, another representation of a tape drive system1300 is shown using, in a reading section, a soft detector 1302, whichprovides automatic compensation for a precoded tape channel 1304. Suchsoft detector 1302 and decoder 404 may be configured the same as thesoft detector 1102 and decoder 404 in FIG. 11. As shown in FIG. 13,however, a first precoder 1306 is positioned before a C1 encoder 1108,such that the output of the first precoder 1306 is sent to the C1encoder 1108. In addition, a second precoder 1308 is positioned afterthe C1 encoder 1108 and immediately prior to a tape channel 202 in theprecoded tape channel 1304, such that the output of the C1 encoder 1108is sent to the second precoder 1308, which, in turn, sends the secondprecoder output to the tape channel 202. Again, the tape channel 202 isshown as a single module representing respective components of the readand the write section, as would be understood by one skilled in the art.

The data which passes through the precoders 1306, 1308 may be describedas having a characteristic of being passed through at least oneprecoder, such that the data may be handled differently and/orcompensated for in a detector which reads a tape which has precoded datarecorded thereon.

One embodiment of the precoded tape channel 1304 is shown in FIG. 14. Asshown, the algorithms provided by the second precoder 1308 and the tapechannel 202 may be combined, as represented by the precoded tape channel1304. Now referring to FIGS. 13-14, bits of data, b, are sent to thesecond precoder 1308. In one embodiment, the precoder 1308 may apply thefollowing equation to b_(k): 1/(1⊕D) where D is delay corresponding tobit duration. The output, α_(k), of the precoder 1308 is then read fromthe tape via the tape channel 202 to generate noiseless signal x_(k),which in this example is represented as an EPR4 signalx_(k)=α_(k)+α_(k−1)−α_(k−2)—α_(k−3). The soft detector 1302 providesautomatic compensation for the presence or absence of a second precoder1308 adjacent the write channel in the writing device by accounting forthe algorithm that the second precoder 1308 applies to the data beforeit is written to the tape and for the transformation that the tapechannel 202 applies to the data as it is read from the tape.

In one embodiment, the first precoder 1306 may apply the followingequation to its input: 1/(1⊕D) where D is repetitive delay.

Furthermore, in one embodiment the soft detector 1302 providingautomatic compensation for the presence of a precoded tape channel 1304may be a noise predictive soft DMAX detector with a noise whiteningfilter. In another approach, the soft detector 1302 providing automaticcompensation for the presence of a precoded tape channel 1304 may be asoft noise-predictive DD-DMAX detector with one or more noise whiteningfilters. In addition, any of these embodiments may be used in a reducedstate detector, as described in more detail herein.

In accordance with one embodiment, the soft detector 1302 providingautomatic compensation for the presence of a precoded tape channel 1304may retain the same number of states and state transitions (e.g.,branches) as in the case where precoding is not used. Additionally, thesoft detector 1302 providing automatic compensation for the presence ofa precoded tape channel 1304 may retain the same branch labeling withnominal output signal x_(k) as in the case where precoding is not used.

As the above examples describe, the soft detector is capable ofproviding automatic compensation for precoded data by selecting aparticular algorithm based on whether or not the precoded data waswritten to the magnetic tape medium by a device having a precoderpositioned immediately adjacent a write channel.

However, is some cases, it is possible to use the same algorithm, suchas a DMAX algorithm, on precoded and non-precoded data. In these cases,the soft detector is capable of providing automatic compensation for theprecoded data by interpreting state transition calculations (detectortrellis branches) based on a type of precoding used in a precoderpositioned immediately adjacent a write channel. In these cases, it ispossible for the soft detector to use a same number of states (memory)whether data was written to the magnetic tape medium via a precoderpositioned immediately adjacent a write channel or not (e.g., the datawas written to the magnetic tape medium without a precoder).

Referring now to FIG. 15, examples of specific precoding cases areshown, according to several embodiments. In trellis structure 1502 onthe left side of FIG. 15, the trellis structure is shown for a casewhere no precoding is used, i.e., b_(k)≡α_(k). As can be seen, thetrellis structure 1502 shows dashed lines for branches where b_(k)≡α_(k)is 0, and solid lines for branches where b_(k)≡α_(k) is 1. For thisexample, an 8-state detector would be used, as the detector has memoryfor α_(k−1), α_(k−2), and α_(k−3) in order to calculate α_(k).

Referring now to trellis structure 1504, in an example using 1/(1⊕D)precoding, it can be seen that the same state transitions(S_(k−1)→S_(k)) occur (branches) as in the non-preceding case, but someof these state transitions are interpreted differently from thenon-precoding example shown as trellis structure 1502. For example, intrellis structure 1504, the branch from 110 to 111 now representsb_(k)=0, instead of α_(k)=1 as in trellis structure 1502.

Similarly, referring to trellis structure 1506, in an example using1/(1⊕D²) precoding, it can be seen that the same state transitions(S_(k−1)→S_(k)) occur (branches) as in the non-precoding case, but someof these state transitions are interpreted differently from thenon-precoding example shown as trellis structure 1502. For example, intrellis structure 1506, the branch from 010 to 001 now representsb_(k)=1, instead of α_(k)=0 as in trellis structure 1502.

It should be noted that the exact same branches occur for each of theprecoded and non-precoded trellis structures. In addition, it should benoted that the same number of states may be used in the detector forcalculating bit estimations regardless of whether the data is precodedor not.

Also, the types of precoding and/or algorithms used in the examplesdescribed herein are not meant to be limiting on what types of precodingand/or algorithms may be used in combination with other embodimentsand/or approaches described herein. Any type of precoding and/oralgorithm known in the art may be used. In addition, the tape channelexamples used herein, EPR4, is not meant to be limiting on what type oftape channel transfer function may be used in conjunction with anyembodiments and/or approaches described herein. Any tape channeltransfer function known in the art may be used. Furthermore, thewhitening filter transfer polynomial examples used herein, are not meantto be limiting on what type of whitening filter transfer function may beused in conjunction with any embodiments and/or approaches describedherein. Any whitening filter transfer function known in the art may beused.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof code, which comprises one or more executable instructions forimplementing the specified logical function(s). It should also be notedthat, in some alternative implementations, the functions noted in theblock may occur out of the order noted in the figures. For example, twoblocks shown in succession may, in fact, be executed substantiallyconcurrently, or the blocks may sometimes be executed in the reverseorder, depending upon the functionality involved. It will also be notedthat each block of the block diagrams and/or flowchart illustration, andcombinations of blocks in the block diagrams and/or flowchartillustration, can be implemented by special purpose hardware-basedsystems that perform the specified functions or acts, or combinations ofspecial purpose hardware and computer instructions.

While various embodiments have been described above, it should beunderstood that they have been presented by way of example only, and notlimitation. Thus, the breadth and scope of an embodiment of the presentinvention should not be limited by any of the above-described exemplaryembodiments, but should be defined only in accordance with the followingclaims and their equivalents.

What is claimed is:
 1. A data storage system, comprising: a softdetector configured to calculate first soft information about each bitof a signal comprising precoded data read from a magnetic tape mediumvia a tape channel, and send the first soft information to a softdecoder, wherein the precoded data comprises a characteristic of beingpassed through at least one precoder prior to being written to themagnetic tape medium, wherein the soft detector provides automaticcompensation for the precoded data, wherein the soft detector applies aDual-Max (DMAX) algorithm comprising:${\alpha_{k}\left( S_{k} \right)} \cong {\max\limits_{S_{k - 1}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\alpha_{k - 1}\left( S_{k - 1} \right)}} \right\}}$${\beta_{k - 1}\left( S_{k - 1} \right)} \cong {\max\limits_{S_{k}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}}$$\begin{matrix}{{{LLR}\left( \alpha_{k} \right)} \cong {{\max\limits_{\underset{a_{k} = {+ 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}} -}} \\{{\max\limits_{\underset{a_{k} = {- 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}},}\end{matrix}$ wherein y_(k) is the signal, α_(k) denotes a bit in a bitsequence of the signal, α_(k)(S_(k)) is an alpha term for a currentstate (S_(k)) in a forward recursion, α_(k)(S_(k−1)) is an alpha termfor a previous state (S_(k−1)) in the forward recursion,β_(k)(S_(k)) isa beta term for the current state in a backward recursion,β_(k−1)(S_(k−1)) is a beta term for the previous state in the backwardrecursion, and LLR(α_(k)) is an approximation of a log-likelihood termthat calculates a posteriori probabilities.
 2. The data storage systemas recited in claim 1, wherein the soft detector provides automaticcompensation for the precoded data by selecting a particular algorithmbased on whether or not the precoded data was written to the magnetictape medium by a device having a precoder positioned immediatelyadjacent a write channel.
 3. The data storage system as recited in claim1, wherein the soft detector uses a same number of states whether datawas written to the magnetic tape medium via a precoder positionedimmediately adjacent a write channel or whether data was written to themagnetic tape medium without a precoder.
 4. The data storage system asrecited in claim 1, wherein the soft detector provides automaticcompensation for the precoded data by interpreting state transitioncalculations based on a type of precoding used in a precoder positionedimmediately adjacent a write channel.
 5. The data storage system asrecited in claim 1, further comprising: a noise whitening filterpositioned preceding the soft detector and configured to pass the signalto the soft detector, wherein the noise whitening filter is configuredto control variance of noise affecting the signal output from thefilter, wherein the noise whitening filter processes the signalaccording to the following transfer polynomial: W(D) =1 −(p₁ D + . . .+p_(λ)D⁸⁰), where p₁ . . . p_(λ) are the noise whitening coefficientswith the noise whitening filter having 2^(λ) states, and where the tapechannel is characterized by a transfer polynomial F(D) =1 +f₁ D + . . .+f_(L) D^(L) where D is delay corresponding to bit duration, with thetape channel having 2 ^(L) states, L represents a memory length of thetape channel, and λ represents a memory length of the noise whiteningfilter.
 6. The data storage system as recited in claim 1, furthercomprising the soft decoder positioned subsequent to the soft detector,the soft decoder being configured to calculate second soft informationabout each bit of the signal and send the second soft information to thesoft detector.
 7. The data storage system as recited in claim 1, whereinthe characteristic of the precoded data is consistent with a precoderthat applies either 1/(1 ⊕D) or 1/(1 ⊕D₂) to bits of data b_(k), where Dis delay corresponding to bit duration.
 8. The data storage system asrecited in claim 1, wherein the DMAX algorithm computes a branch metric,m_(k)(S_(k−1), S_(k)), of a transition to a current state (S_(k)) from aprevious state (S_(k−1)) represented by: m_(k)(S_(k−1),S_(k))=−(y_(k)−x_(k))²/(2σ²)+ln P(b_(k)), where σ² is prediction noisevariance, P(b_(k)) are a priori probabilities provided by the softdecoder, y_(k) is an input sample, and x_(k) is an ideal nominal outputsignal from the tape channel.
 9. The data storage system as recited inclaim 8, wherein the DMAX algorithm is normalized according to thefollowing relationship:${\alpha_{k}\left( S_{k} \right)} = {\max\limits_{S_{k - 1}}{\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)}} \right\}\mspace{14mu}\left( {{k = 1},\ldots\mspace{14mu},N} \right)}}$${{{Normalization}\mspace{11mu}\text{:}\mspace{14mu}{find}\mspace{14mu} A_{k}} = {\max\limits_{S_{k}}\left\{ {\alpha_{k}\left( S_{k} \right)} \right\}}},{{{replace}\mspace{14mu}{\alpha_{k}\left( S_{k} \right)}}->{{\alpha_{k}\left( S_{k} \right)} - {A_{k}\mspace{14mu}{\forall{\alpha_{k}\left( S_{k} \right)}}}}}$${\beta_{k - 1}\left( S_{k - 1} \right)} = {\max\limits_{S_{k}}{\left\{ {{\beta_{k}\left( S_{k} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)}} \right\}\mspace{14mu}\left( {{k = N},\ldots\mspace{14mu},2} \right)}}$${{{Normalization}\mspace{11mu}\text{:}\mspace{14mu}{find}\mspace{14mu} B_{k - 1}} = {\max\limits_{S_{k - 1}}\left\{ {\beta_{k - 1}\left( S_{k - 1} \right)} \right\}}},{{{replace}\mspace{14mu}{\beta_{k - 1}\left( S_{k - 1} \right)}}->{{\beta_{k - 1}\left( S_{k - 1} \right)} - {B_{k - 1}\mspace{14mu}{\forall{\beta_{k - 1}\left( S_{k - 1} \right)}}}}},$wherein A_(k) denotes a maximum value for α_(k) across all states(S_(k)), and wherein α_(k) for each state (S_(k)) is replaced with avalue equaling α_(k) minus A_(k) for normalization, and whereinβ_(k−1)denotes a maximum value for , β_(k−1)across all previous states(S_(k)), and wherein,8k for each previous state (S_(k)) is replaced witha value equaling , β_(k−1)minus B_(k−1)for normalization.
 10. A method,comprising: reading precoded data from a magnetic tape medium to producea signal; passing the signal through a noise whitening filter, whereinthe noise whitening filter is configured to control variance of noiseaffecting the signal output from the filter, wherein the noise whiteningfilter processes the signal according to the following transferpolynomial: W(D)=1−(p₁D+ . . . +p_(λ)D^(λ)), where p₁ . . . p_(λ) arethe noise whitening coefficients with the noise whitening filter having2^(λ) states, and where the tape channel is characterized by a transferpolynomial F(D)=1+f₁D+ . . . +f_(L)D^(L) where D is delay correspondingto bit duration, with the tape channel having 2^(L) states, L representsa memory length of the tape channel, and λ represents a memory length ofthe noise whitening filter, passing the signal through a soft detectorto calculate first soft information about each bit of the signal and toprovide adaptive compensation for the precoded data; sending the firstsoft information to a soft decoder, passing the signal through the softdecoder to calculate second soft information about each bit of thesignal; and sending the second soft information to the soft detector,wherein the precoded data comprises a characteristic of being passedthrough at least one precoder prior to being written to the magnetictape medium.
 11. The method as recited in claim 10, wherein the softdetector provides automatic compensation for the precoded data byselecting a particular algorithm based on whether or not the precodeddata was written to the magnetic tape medium by a device having aprecoder positioned immediately adjacent a write channel.
 12. The methodas recited in claim 10, wherein the soft detector uses a same number ofstates whether data was written to the magnetic tape medium via aprecoder positioned immediately adjacent a write channel or whether datawas written to the magnetic tape medium without a precoder.
 13. Themethod as recited in claim 10, wherein the soft detector providesautomatic compensation for the precoded data by interpreting statetransition calculations based on a type of precoding used in a precoderpositioned immediately adjacent a write channel.
 14. The method asrecited in claim 10, wherein the whitening filter coefficients p₁, p₂, .. . , p_(λ) are estimated adaptively.
 15. The method as recited in claim10, wherein the soft detector applies a Dual-Max (DMAX) algorithmcomprising:${\alpha_{k}\left( S_{k} \right)} \cong {\max\limits_{S_{k - 1}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\alpha_{k - 1}\left( S_{k - 1} \right)}} \right\}}$${\beta_{k - 1}\left( S_{k - 1} \right)} \cong {\max\limits_{S_{k}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}}$$\begin{matrix}{{{LLR}\left( \alpha_{k} \right)} \cong {{\max\limits_{\underset{a_{k} = {+ 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}} -}} \\{{\max\limits_{\underset{a_{k} = {- 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}},}\end{matrix}$ wherein y_(k) is the signal, α_(k) denotes a bit in a bitsequence of the signal, α_(k)(S_(k)) is an alpha term for a currentstate (S_(k)) in a forward recursion, α_(k−1)(S_(k−1)) is an alpha termfor a previous state (S_(k−1)) in the forward recursion β_(k)(S_(k)) isa beta term for the current state in a backward recursion,β_(k−1)(S_(k−1)) is a beta term for the previous state in the backwardrecursion, and LLR(α_(k)) is an approximation of a log-likelihood termthat calculates a posteriori probabilities.
 16. The method as recited inclaim 15, wherein the characteristic of the precoded datais consistentwith a precoder that applies either 1/(1⊕D) or 1/ (1⊕D²) to bits of datab_(k),where D is delay corresponding to bit duration.
 17. The method asrecited in claim 16, wherein the DMAX algorithm computes a branchmetric, m_(k)(S_(k−1), S_(k)), of a transition to a current state(S_(k)) from a previous state (S_(k−1))represented by: m_(k)(S_(k−1),S_(k))=−(y_(k)−x_(k))²/(2σ²)+ln P(b_(k)), where σ² is prediction noisevariance, P(b_(k)) are a priori probabilities provided by the softdecoder, y_(k) is an input sample, and x_(k) is an ideal nominal outputfrom the tape channel.
 18. The method as recited in claim 17, whereinthe DMAX algorithm is normalized according to the followingrelationship:${\alpha_{k}\left( S_{k} \right)} = {\max\limits_{S_{k - 1}}{\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)}} \right\}\mspace{14mu}\left( {{k = 1},\ldots\mspace{14mu},N} \right)}}$${{{Normalization}\mspace{11mu}\text{:}\mspace{14mu}{find}\mspace{14mu} A_{k}} = {\max\limits_{S_{k}}\left\{ {\alpha_{k}\left( S_{k} \right)} \right\}}},{{{replace}\mspace{14mu}{\alpha_{k}\left( S_{k} \right)}}->{{\alpha_{k}\left( S_{k} \right)} - {A_{k}\mspace{14mu}{\forall{\alpha_{k}\left( S_{k} \right)}}}}}$${\beta_{k - 1}\left( S_{k - 1} \right)} = {\max\limits_{S_{k}}{\left\{ {{\beta_{k}\left( S_{k} \right)} + {m_{k}\left( {S_{k - 1},S_{k}} \right)}} \right\}\mspace{14mu}\left( {{k = N},\ldots\mspace{14mu},2} \right)}}$${{{Normalization}\mspace{11mu}\text{:}\mspace{14mu}{find}\mspace{14mu} B_{k - 1}} = {\max\limits_{S_{k - 1}}\left\{ {\beta_{k - 1}\left( S_{k - 1} \right)} \right\}}},{{{replace}\mspace{14mu}{\beta_{k - 1}\left( S_{k - 1} \right)}}->{{\beta_{k - 1}\left( S_{k - 1} \right)} - {B_{k - 1}\mspace{14mu}{\forall{\beta_{k - 1}\left( S_{k - 1} \right)}}}}},$wherein A_(k) denotes a maximum value for α_(k) across all states(S_(k)), and wherein α_(k) for each state (S_(k)) is replaced with avalue equaling α_(k) minus A_(k) for normalization, and wherein B_(k−1)denotes a maximum value for β_(k−1) across all previous states(S_(k−1)), and wherein β_(k) for each previous state (S_(k−1)) isreplaced with a value equaling β_(k−1), minus B_(k−1) for normalization.19. A computer program product, comprising a computer readable storagemedium having computer readable program code embodied therewith, thecomputer readable program code readable/executable by a processor tocause the processor to: receive, by the processor, a signal comprisingprecoded data read from a magnetic tape medium and pass the signalthrough a soft detector to calculate first soft information about eachbit of the signal and to provide adaptive compensation for the precodeddata; send, by the processor, the first soft information to a softdecoder; pass, by the processor, the signal through the soft decoder tocalculate second soft information about each bit of the signal; andsend, by the processor, the second soft information to the softdetector, wherein the precoded data comprises a characteristic of beingpassed through at least one precoder prior to being written to themagnetic tape medium, and wherein the characteristic of the precodeddata is consistent with a precoder that applies 1/(1⊕D²) to bits of datab_(k), where D is delay corresponding to bit duration.
 20. The computerprogram product as recited in claim 19, wherein the computer readableprogram code is further readable/executable by the processor to pass, bythe processor, the signal through a noise whitening filter to minimizevariance of the noise affecting the signal output from the filter,wherein the noise whitening filter is configured to process the signalaccording to the following transfer polynomial: W(D)=1−(p₁D+, where p₁.. . p_(λ) are noise whitening coefficients with the noise whiteningfilter having 2/^(λ)states, and where the tape channel is characterizedby a transfer polynomial F(D) =1 +f₁ D +. . . +f_(L) D^(L) where D isdelay corresponding to bit duration, with the tape channel having 2 ^(L)states, L represents a memory length of the tape channel, andλrepresents a memory length of the noise whitening filter, wherein thesoft detector applies a Dual-Max (DMAX) algorithm comprising:${\alpha_{k}\left( S_{k} \right)} \cong {\max\limits_{S_{k - 1}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\alpha_{k - 1}\left( S_{k - 1} \right)}} \right\}}$${\beta_{k - 1}\left( S_{k - 1} \right)} \cong {\max\limits_{S_{k}}\left\{ {{\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}}$$\begin{matrix}{{{LLR}\left( \alpha_{k} \right)} \cong {{\max\limits_{\underset{a_{k} = {+ 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}} -}} \\{{\max\limits_{\underset{a_{k} = {- 1}}{S_{k - 1}->S_{k}}}\left\{ {{\alpha_{k - 1}\left( S_{k - 1} \right)} + {\gamma_{k}\left( {S_{k - 1},S_{k}} \right)} + {\beta_{k}\left( S_{k} \right)}} \right\}},}\end{matrix}$ and wherein y_(k) is the signal, α_(k) denotes a bit in abit sequence of the signal, α_(k)(S_(k)) is analpha term for a currentstate (S_(k)) in a forward recursion, α_(k)(S_(k−1)) is an alpha termfor aprevious state (S_(k−1)) in the forward recursion β_(k)(S_(k)) is abeta term for the current state in abackward recursion, β_(k−1)(S_(k−1))is a beta term for the previous state in the backward recursion, andLLR(α_(k)) is an approximation of a log-likelihood term that calculatesa posteriori probabilities.